Physics 780.20: Assignment #1

Here are some hints, suggestions, and comments on the assignment.

Summing up vs. summing down

You should find it useful to think of how the 1 + a + a + ... vs.
a + a + ... + 1 problem from Session 2 relates to this one. (That
might also be a good code to start from in writing one for this
problem.)

If you only consider relatively small N, you will not see any
difference between summing up and summing down, since both will have an
error about equal to the machine precision. So be sure to increase N
until you see a different trend (I used up to 10^8).

If you are looking for the effects of errors over a logarithmic
range (e.g., N from 10^2 to 10^8), you should vary N by multiplying
by a factor each time through a loop and not by adding 1 (or some other
number). That is, don't use:

for (N = 100; N <= 100000000; N++)
{
}

which will take an enormous amount of time calculating lots of N values
you don't need. Instead, use something like:

for (N = 100; N <= 100000000; N *= 10)
{
}

which just calculates for 100, 1000, 100000, etc.

When you output relative errors to a file for plotting,
the "scientific" format is usually most useful,
again because you are looking at logaithmic intervals.
(See the codes we've used in class for examples.)

It is only the absolute value of the relative error that is of
interest. So use "fabs" (you need to "input ").
It is particularly important to have only
positive numbers if you are going to plot on logarithmic scales!

Be careful that you don't confuse the machine precision with the
smallest floating point number. For example,
in single precision, the machine
precision is about 10-7, but that does not
mean that 10^-10 is set to zero. It is only when you add
two numbers that differ by a factor of more than
10-7 that the smaller number is effectively zero.

A common bug when you have an outer loop stepping through
N_max and an inner loop that does the sum of 1/n up to N_max
is forgetting to reset your sum to zero when you start the
inner loop. So if your summation variable is sum_up,
make sure that sum_up = 0.; appears within the outer
loop and not just at the beginning of the program.

Bessel Function Recursing Up and Down

You will just be modifying a code from Session 2, not writing
one from scratch.

The interpretation of the graph should mainly explain the
meaning of the regions
where the plotted error is about unity and where it is very small.
(I.e., which result is good and which is poor, or are both good
or poor?)

For part c), remember that you need
the linker option -lgsl -lgslcblas in your makefile
to run a program that uses the GSL library.
Don't forget that you need to #include a header file
(see the Bessel function example from Session 1).